We analyze entropy solutions for a class of Levy mixed hyperbolicparabolic equations containing a non-local (or fractional) diffusion operator originating from a pure jump Levy process. For these solutions we establish uniqueness (L1 contraction property) and continuous dependence results.
In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative Lévy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.
Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time dependent coercive operators. We consider abstract Galerkin approximations in space combined with theta-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular the results apply to partial integro differentia…
The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Levy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.
We study a hyperbolic-parabolic model of chemotaxis in dimensions one and two. In particular, we prove the global existence of classical solutions in certain dissipation regimes.
In this small paper, we study a boundary value problem for an equation of parabolic-hyperbolic type. The goal is to show how we can prove existence and uniqueness theorem for a regular solution.
We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes.
The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.
We discuss the well-posedness and decay of Besicovitch almost periodic solutions for a class of nonlinear degenerate anisotropic hyperbolic-parabolic equations. In our definition of weak entropy solution the initial data is only assumed in a weak sense, and in this connection the strong trace of the solution in the initial time hyperplane is also established.
We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria \textit{Escherichia Coli}. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prov…
We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosi…
The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic (MHD) model in two dimensional space. Based on Agmon, Douglis and Nirenberg's estimates for the stationary Stokes equation and the Solonnikov's theorem of $L^p$-$L^q$-estimates for the evolution Stokes equation, it is shown that the mixed-type MHD equations exist a global strong solution.
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field ar…
This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of the same type as the problem with oscillating coefficients.
Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise
Centre for Applicable Mathematics, Tata Institute of Fundamental Research（应用数学中心，塔塔基础研究所）；Mathematisches Institut, Universität Tübingen（蒂宾根大学数学研究所）；LMAP UMR- CNRS 5142（LMAP UMR-CNRS 5142）
We prove the well-posedness and the asymptotic decay to the mean value of Besicovitch almost periodic entropy solutions to nonlinear aniso\-tropic degenerate parabolic-hyperbolic equations. After setting up the problem and its kinetic formulation on the Bohr compact, the main result, that is, the decay property, ia achieved by devising a suitable adaptation of the technique introduced by Chen and Perthame (2009) in their proof of the decay of periodic entropy solutions to the same equations.
In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and non- homogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.
We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions.
Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying Hörmander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable, possibly nonsymmetric, Levy measure. Some rough probability density function estimates of the associated Levy process are used as well.
We are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and Dirichlet boundary data. Moreover, we prove a $L^1-$type contraction property for weak entropy solutions obtained via parabolic perturbation. This is a weak selection principle which means that the weak entropy solutions are stable in this class.
In this paper, we derive the large-time profile of solutions to the Cauchy problem of a hyperbolic-parabolic system modeling the vasculogenesis in $\R^3$. When the initial data are prescribed in the vicinity of a constant ground state, by constructing a time-frequency Lyapunov functional and employing the Fourier energy method and spectral analysis, we show that solution of the Cauchy problem tend time-asymptotically to linear diffusion waves around the constant ground state with algebraic decaying rates under certain conditions on the density-dependent pressure function.
We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a gener…
Stochastic parabolic integro-differential problem is considered in the whole space. By verifying Hörmander condition, the existence and uniqueness is proved in Lp-spaces of functions whose regularity is defined by a scalable Levy measure. Some rough probability density function estimates of the associated Levy process are used as well.
We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecti…
We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain $\cO=\cO'\X\cO''$ where a Neumann boundary condition is imposed on $\po\cO'\X\cO"$, the hyperbolic boundary, and a Dirichlet condition is imposed on $\cO'\X\po\cO"$, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic …
We develop a time and space dependent predator-prey model. The predators' equation is a non local hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators "hunt" for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey-parasites. The general well posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilae used to fight aga…
An Euler-type hyperbolic-parabolic system of chemotactic aggregation describing the vascular network formation is investigated in the critical regularity setting. For small initial data around a constant equilibrium state, the well-posedness of the global classical solution to the Cauchy problem with general pressure laws is established in homogeneous hybrid Besov spaces. Then, the optimal time-decay rates of the global solution are analyzed under an additional regularity assumption on the initial data. Furthermore, the relaxation limit (large friction limit) of the hyperbolic-parabolic system…
In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis in $\mathbb{R}$. Under some suitable structural assumption on the pressure function, we first predict the system admits a nonlinear diffusion wave in $\mathbb{R}$ based on the empirical results in the literature. Then we show that the solution of the concerned system will locally and asymptotically converges to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergen…
This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)ΔU(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive. Under the behavior of the diffusion coefficient $D$ with polynomial order at spatial infinity, a family of supersolutions with slowly decaying property at spatial infinity is provided. As a first application, weighted $L^2$ type decay estimates for the initial-boundary value problem of the corresponding parabolic equation are proved. The second application is the …
We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.
